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Let dF denote the force acting on the area, on the upper base, of a ring of radius r and width dr, then the stress
But if 6 is the angle of twist at the upper
base and I the length of the cylinder, then the strain equals, rd_
I '
Therefore by Hooke's law dF       , rd
2-irrdr        I
In this case X is called modulus of shearing elasticity or, simply, shear modulus. Solving the preceding equation for dF we get
Therefore the torque acting upon the area of the ring is
do = r  dF
O       A
0r3 dr.
2?rX I
9 P
J (\
61  rzdr
= X
FIG. 97.
where G is the total torque applied at the upper end and a the radius of the cylinder. Thus the torque necessary to produce a given angle of twist varies directly as the fourth power of the radius and inversely as the length. On the other hand for a given shaft the torque varies directly with the angle of twist.
The torsional rigidity of the shaft is defined as the torque necessary to produce a unit angular twist; therefore